Random Strict Partitions and Determinantal Point Processes 167
Remark 1. Note that the kernel K
ν,ξ
given by 8 can be viewed as a discrete analogue of an integrable operator if as variables we take x
2
and y
2
. About integrable operators, e.g., see [16, 11]. Discrete integrable operators are discussed in [3] and [7, §6].
Remark 2. There is an identity
φ x =
e φx
1 + ξ −
ξ1 + 2ν − x1 − ξ x1
− ξ
2
φ
1
x, 9
which is a combination of 2.838, 2.839 and 2.92 in [12]. Therefore, A
2
x = A
1
x + c Bx, where c does not depend on x. Thus, the kernel 8 with A
1
is identical to the one with A
2
. Furthermore, all our formulas must be symmetric with respect to the replacement of ν by
−ν. Clearly, Ξx, y and all the functions φ
i
x, i ∈ Z
≥0
, possess this property, so the kernel 7 and the kernel 8 with A
1
do not change under the substitution ν → −ν. The same holds for the
kernel 8 with A
2
, because from 9 we have e φx|
ν→−ν
= e φx +
ec x
φ
1
x, where ec does not
depend on x.
2.3 Scheme of proof of Theorems 2.1 and 2.2
We begin with the argument similar to [26], but instead of the infinite wedge space we take the Fock space with the orthonormal basis v
λ
= e
λ
1
∧ e
λ
2
∧ · · · ∧ e
λ
ℓλ
indexed by all strict partitions in particular, v
;
= 1. A similar space is used in [24, §3] and [38, §5.2]. By calculations in this Fock space we first obtain a Pfaffian formula for the correlation functions ρ
ν,ξ
of the point process P
ν,ξ
and not a determinantal formula as it was in [26].
Proposition 2. There exists a function Φ
ν,ξ
: Z \ {0}
2
→ C such that for every finite subset X = x
1
, . . . , x
n
⊂ Z we have
ρ
ν,ξ
X = −1
P
n i=1
x
i
· Pf ΦX ,
where Pf means Pfaffian. Here ΦX is the 2n × 2n skew-symmetric matrix with rows and columns
indexed by x
1
, . . . , x
n
, −x
n
, . . . , −x
1
such that the i j-th element of the matrix ΦX above the main diagonal is Φ
ν,ξ
i, j, where i and j take values x
1
, . . . , x
n
, −x
n
, . . . , −x
1
. Now we explain how one can convert the above Pfaffian formula for the correlation functions of
P
ν,ξ
to a determinantal one. It turns out that Φ
ν,ξ
satisfies the following identities here x, y ∈
Z \ {0}:
• If x 6= y, then Φ
ν,ξ
x, − y = −1
y x+ y x
− y
Φ
ν,ξ
x, y. • If x 6= − y, then Φ
ν,ξ
y, x = −Φ
ν,ξ
x, y and, moreover, Φ
ν,ξ
−x, − y = −1
x+ y+1
Φ
ν,ξ
x, y. • If x 6= 0, then Φ
ν,ξ
x, −x + Φ
ν,ξ
−x, x = −1
x
. Fix a finite subset X =
x
1
, . . . , x
n
⊂ Z . Define C
kl
:= δ
kl
+ −1
x
k ∧l
x
k ∧l
−x
n
x
k ∧l
+x
n
I
{k+l=2n+1}
k, l = 1, . . . , 2n, where k
∧ l means the minimum of k and l, and I means the indicator. Clearly, the 2n
× 2n matrix C = [C
kl
] is invertible. Using the above identities for Φ
ν,ξ
, we obtain CΦX C
′
= M
−M
′
, where ..
′
means the matrix transpose and M has format n × n. It follows from properties of
Pfaffians that Pf ΦX = −1
nn −12
det C
−1
det M . There exist two diagonal n × n matrices D
1
168 Electronic Communications in Probability
and D
2
such that detD
1
D
2
= −1
P
n i=1
x
i
det C
−1
and D
1
M
D
2
= K
ν,ξ
X = [K
ν,ξ
x
i
, x
j
]
n i, j=1
for some Z × Z
matrix K
ν,ξ
. Here M
is the matrix that is obtained from M by rotation by 90 degrees counter–clockwise. Note that det M
= −1
nn −12
det M . Thus, ρ
ν,ξ
X = −1
P
n i=1
x
i
Pf ΦX = det K
ν,ξ
X , which means that K
ν,ξ
is the desired correla- tion kernel. The kernel K
ν,ξ
is related to Φ
ν,ξ
as follows: K
ν,ξ
x, y = 2
−1
y
p x y
x + y Φ
ν,ξ
x, − y, x, y
∈ Z .
10 We obtain explicit expressions for Φ
ν,ξ
in terms of the Gauss hypergeometric function. Their form is similar to formulas 3.16 and 3.17 in [26]. These expressions for Φ
ν,ξ
together with relation 10 imply Theorems 2.1 and 2.2, respectively.
Remark 3. Let L
ν,ξ
be the operator defined by 5 with ψ = ψ
ν,ξ
given by 4. Once formula 8 for K
ν,ξ
is obtained, one can directly check that K
ν,ξ
= L
ν,ξ
1 + L
ν,ξ −1
. Indeed, this relation is equivalent to K
ν,ξ
+ K
ν,ξ
L
ν,ξ
− L
ν,ξ
= 0, and the computation of the matrix product K
ν,ξ
L
ν,ξ
mainly reduces to the computation of sums of the form P
∞ k=1
ξ
k
Γ
1 2
+ν+kΓ
1 2
−ν+k kk
−1 f k
k+a
, where a 6=
−1, −2, . . . is some constant and f k is one of the functions kφ k, kφ
1
k, or φ
1
k. These sums can be computed using Lemma 3.4 in Appendix in [7].
2.4 Double contour integral representations
